Skip to main content

Full text of "Men Of Mathematics"

See other formats

obviousness of this simple assumed equality, V2 x Vs = y^
taken for granted in school arithmetic, is evident if we visualize
what the equality implies: the 'lawless' square roots of 2, 3, Q
are to be extracted, the first two of these are then to be multi-
plied together, and the result is to come out equal to thte third,
As not one of these three roots can be extracted exactly > no
matter to how many decimal places the computation is carried,
it is clear that the verification by multiplication as just de-
scribed will never be complete. The whole human race toiling
incessantly through all its existence could never prove in this
way that ,V2 X Vs = V6. Closer and closer approximations
to equality would be attained as time went on, but finality
would continue to recede. To make these concepts of 'approxi-
mation' and 'equality' precise, or to replace our first crude
conceptions of irrationals by sharper descriptions which -mil
obviate the difficulties indicated, was the task Dedekind set
himself in the early 1870's - his work on Continuity and
Irrational Numbers was published in 1872.
The heart of Dedeldnd's theory of irrational numbers is his
concept of the 'cut1 or 'section' (Schnitt): a cut separates oB
rational numbers into two classes, so that each number in the
first class is less than each number in the second class; every such
cut which does not 'correspond' to a rational number 'defines*
an irrational number. This bald statement needs elaboration,
particularly as even an accurate exposition conceals certain
subtle difficulties rooted in the theory of the mathematical
infinite, which will reappear when we consider the life of
Dedekind's friend Cantor.
Assume that some rule has been prescribed which separates
all rational numbers into ftao classes, say an 'upper' class and a
'lower' class, such that each number in the lower class is less than
every number in the upper class. (Such an assumption would
not pass unchallenged to-day by all schools of mathematical
philosophy. However, for the moment, it may be regarded as
unobjectionable.) On this assumption one of three mutually
exclusive situations is possible.
(A) There may be a number in the lower class which is greatet
than every other number in that class.